Galois Representations and Modular Forms

Galois representations and modular forms

Takeshi Saito July 17-22, 2006 at IHES

Abstract This note is based on a series of lectures given at the summer school held on July 17-29, 2006 at IHES. The purpose of the lectures is to explain the basic ideas in the geometric construction of the Galois representations associated to elliptic modular forms of weight at least 2.

Motivation

The Galois representations associated to modular forms play a central role in the modern number theory. In this introduction, we give a reason why they take such a position. A goal in number theory is to understand the ﬁnite extensions of Q. By Galois Q Q

theory, it is equivalent to understand the absolute Galois group GÉ = Gal( / ). One may say that one knows a group if one knows its representations. Representations are classiﬁed by the degrees. The class ﬁeld theory provides us a precise understanding of the representations of degree 1, or characters. By the theorem → C×

of Kronecker-Weber, a continuous character GÉ is a Dirichlet character → Q Q → Z Z × → C×

GÉ Gal( (ζN )/ ) ( /N ) for some integer N ≥ 1. If we consider not only complex continuous characters but → Q× also -adic characters GÉ for a prime , we ﬁnd more characters. For example, the -adic cyclotomic character is deﬁned as the composition:

n × × × → Q n ∈ N Q Q n Q → Z Z Z ⊂ Q GÉ Gal( (ζ ,n )/ ) =← lim− nGal( (ζ )/ ) ←lim− n( / ) = . The -adic characters “with motivic origin” are generated by Dirichlet characters and -adic cyclotomic characters: { }

“geometric” -adic character of GÉ = Dirichlet characters, -adic cyclotomic characters if we use a fancy terminology “geometric”, that will not be explained in this note. For the deﬁnition, we refer to [13].

1 When we leave the realm of class ﬁeld theory, the ﬁrst representations we encounter are those of degree 2. For -adic Galois representation of degree 2, we expect to have (cf. [13]) a similar equality

{ }

odd “geometric” -adic representation of GÉ of degree 2 of distinct Hodge-Tate weight = {-adic representation associated to modular form of weight at least 2}, up to twist by a power of the cyclotomic character. In other words, the Galois representations associated to modular forms are the first ones we encounter when we explore outside the domain of class field theory. In this note, we discuss only one direction ⊃ established by Shimura and Deligne ([21], [7]). We will not discuss the other direction ⊂, which is almost established after the revolutionary work of Wiles, although it has significant consequences including Fermat’s last theorem, the modularity of elliptic curves, etc. ([24], [1]). In Section 1, we recall the definition of modular forms and state the existence of Galois representations associated to normalized eigen cusp forms. We introduce C Z 1 modular curves defined over and over [ N ] as the key ingredient in the construction of the Galois representations, in Section 2. Then, we construct the Galois representations in the case of weight 2 by decomposing the Tate module of the Jacobian of a modular curve in Section 3. In the final Section 4, we briefly sketch an outline of the construction in the higher weight case. Proofs will be only sketched or omitted mostly. The author apologizes that he also omits the historical accounts completely. The author would like to thank the participants of the summer school for pointing out numerous mistakes and inaccuracies during the lectures.

Contents

1 Galois representations and modular forms 3 1.1 Modular forms ...... 3 1.2 Examples ...... 4 1.3 Hecke operators ...... 6 1.4 Galois representations ...... 7

2 Modular curves and modular forms 8 2.1 Elliptic curves ...... 8 2.2 Elliptic curves over C ...... 10 2.3 Modular curves over C ...... 11 2.4 Modular curves and modular forms ...... 14 Z 1 2.5 Modular curves over [ N ] ...... 15 2.6 Hecke operators ...... 17

2 3 Construction of Galois representations: the case k =2 19 3.1 Galois representations and ﬁnite ´etale group schemes ...... 19 3.2 Jacobian of a curve and its Tate module ...... 19 3.3 Construction of Galois representations ...... 22 3.4 Congruence relation ...... 23

4 Construction of Galois representations: the case k>2 25 4.1 Etale cohomology ...... 25 4.2 Construction of Galois representations ...... 27

References 27

1 Galois representations and modular forms

1.1 Modular forms Let N ≥ 1 and k ≥ 2 be integers and ε :(Z/N Z)× → C× be a character. We will define C-vector spaces Sk(N,ε) ⊂ Mk(N,ε) of cusp forms and of modular forms of level N, weight k and of character ε. We will see later in §3.4 that they are of finite dimension by using the compactification of a modular curve. For ε = 1, we write Sk(N) ⊂ Mk(N) for Sk(N,1) ⊂ Mk(N,1). For a reference on this subsection, we refer to [12] Chapter 1. A subgroup Γ ⊂ SL2(Z) is called a congruence subgroup if there exists an integer N ≥ 1 such that Γ ⊃ Γ(N) = Ker(SL2(Z) → SL2(Z/N Z)). In this note, we mainly consider the congruence subgroups ab Γ (N)= ∈ SL (Z) a ≡ 1,c≡ 0modN 1 cd 2 ab ⊂ Γ (N)= ∈ SL (Z) c ≡ 0modN . 0 cd 2 ab We identify the quotient Γ (N)/Γ (N) with (Z/N Z)× by → d mod N. The 0 1 cd indices are given by 1 [SL (Z):Γ(N)] = (p +1)pordp(N)−1 = N 1+ , 2 0 p p|N p|N 1 [SL (Z):Γ(N)] = (p2 − 1)p2(ordp(N)−1) = N 2 1 − . 2 1 p2 p|N p|N

The action of SL2(Z) on the Poincar´e upper half plane H = {τ ∈ C|Im τ>0} is deﬁned by aτ + b γ(τ)= cτ + d

3 ab for γ = ∈ SL (Z) and τ ∈ H. For a holomorphic function f on H, we deﬁne cd 2 ∗ a holomorphic function γkf on H by 1 γ∗f(τ)= f(γτ). k (cτ + d)k

∗ ∗ If k = 2, we have γ (fdτ)=γ2 (f)dτ.

Definition 1.1 Let Γ ⊃ Γ(N) be a congruence subgroup and k ≥ 2 be an integer. We say that a holomorphic function f : H → C is a modular form (resp. a cusp form) of weight k with respect to Γ, if the following conditions (1) and (2) are satisfied. ∗ ∈ (1) γkf = f for all γ Γ. ∗ ∗ ∗ (2) For each γ ∈ SL2(Z), γ f satisfies γ f(τ + N)=γ f(τ) and hence we have k ∞ k k ∗ n ∗ n τ a Fourier expansion γkf(τ)= n=−∞ a N (γkf)qN where qN = exp(2πiN ). We require the condition ∗ n a N (γkf)=0 be satisfied for n<0 (resp. n ≤ 0) for every γ ∈ SL2(Z).

We put { | }

Sk(Γ) = f f is a cusp form of weight k w.r.t. Γ ⊂ { | }

Mk(Γ) = f f is a modular form of weight k w.r.t. Γ and deﬁne Sk(N)=Sk(Γ0(N)). Since Γ0(N) contains Γ1(N) as a normal subgroup, the → ∗ group Γ0(N) has a natural action on Sk(Γ1(N)) by f γkf. Since Γ1(N) acts trivially × on Sk(Γ1(N)), we have an induced action of the quotient Γ0(N)/Γ1(N)=(Z/N Z) on × Sk(Γ1(N)). The action of d ∈ (Z/N Z) on Sk(Γ1(N)) is denoted by d and is called the diamond operator. The space Sk(Γ1(N)) is decomposed by the characters Sk(Γ1(N)) = Sk(N,ε)

× ×

→ ε:(/N )

× where Sk(N,ε)={f ∈ Sk(Γ1(N))|df = ε(d)f for all d ∈ (Z/N Z) }. The ﬁxed part Γ0(N) Sk(Γ1(N)) = Sk(N,1) is equal to Sk(N)=Sk(Γ0(N)).

1.2 Examples We give some basic examples following [19] Chapter VII. First, we deﬁne the Eisenstein series. For an even integer k ≥ 4, we put 1 Gk(τ)= k . (mτ + n) m,n∈,(m,n)=(0,0)

It is a modular form of weight k.

4 The q-expansion of an Eisenstein series is computed as follows. The logarithmic ∞ τ 2 derivative of sin πτ = πτ 1 − gives n2 n=1 ∞ ∞ 1 1 1 1 −2πi + qn = + + . 2 τ τ + n τ − n n=1 n=1 d 1 d Applying k − 1-times the operator q = , one gets dq 2πi dτ ∞ (−1)k(k − 1)! 1 nk−1qn = . (2πi)k (τ + n)k

n=1 n∈ ≥ k−1 For k 4 even, by putting σk−1(n)= d|n d and ∞ 2 n E (q)=1+ σ − (n)q ∈ Q[[q]], k ζ(1 − k) k 1 n=1 we deduce (k − 1)! (k − 1)! G (τ)= (2ζ(k)+(G (τ) − 2ζ(k))) (2πi)k k (2πi)k k ∞ n = ζ(1 − k)+2 σk−1(n)q = ζ(1 − k)Ek(q). n=1 Recall that the special values of the Riemann zeta function at negative odd integers 1 1 1 ζ(−1) = − ,ζ(−3) = ,ζ(−5) = − , ... 12 120 252 are non-zero rational numbers (cf. [19] p.71, Chapter VII Proposition 7). The C-algebra ∞ of modular forms of level 1 are generated by the Eisenstein series: k=0 Mk(1) = C[E4,E6] (cf. loc. cit. Section 3.2). The Delta-function deﬁned by ∞ ∞ 1 Δ(q)= (E3 − E2)=q (1 − qn)24 = τ(n)qn 123 4 6 n=1 n=1 is a cusp form of weight 12 and of level 1 (see [19] Chapter VII Sections 4.4, 4.5). The space of cusp forms of level 1 are generated by the Delta-function as a module over the ∞ C · algebra of modular forms: k=0 Sk(1) = [E4,E6] Δ. The function f11 deﬁned by ∞ n 2 11n 2 f11(q)=q (1 − q ) (1 − q ) n=1

is a basis of the space of cusp forms S2(11) of level 11 and of weight 2 (see [12] Proposition 3.2.2).

5 1.3 Hecke operators The space of modular forms has Hecke operators as its endomorphisms. More detail on Hecke opeators can be found in [12] Chapter 5. For every integer n ≥ 1, the Hecke operator Tn is deﬁned as an endomorphism of Sk(Γ1(N)). Here we only consider the case where n = p is a prime. The general case is discussed later in §2.6. For a prime number p, we deﬁne the Hecke operator Tp by p−1 1 τ + i pk−1pf(pτ)ifp N T f(τ)= f + (1) p p p | i=0 0ifp N. n In terms of the q-expansion f(τ)= n an(f)q , we have k−1 pn n/p p n an( p f)q if p N Tpf(τ)= an(f)q + 0ifp|N. p|n

The Hecke operators on Sk(Γ1(N)) are commutative to each other and formally satisfy the relation ∞ −s −s k−1 −2s −1 −s −1 Tnn = (1 − Tpp + pp p ) × (1 − Tpp ) . | n=1 p¹N p N ∞ n ∈ A cusp form f = n=1 anq Sk(N,ε) is called a normalized eigenform if Tnf = ≥ ∈ λnf for all n 1 and if a1 = 1. Since a1(Tnf)=an(f), if f Sk(N,ε) is a normalized ∞ n eigenform, we have λn = an. For a normalized eigenform f = n=1 anq , the subfield Q(f)=Q(an,n ∈ N) ⊂ C is a finite extension of Q, as we will see later at the end of §2.6. Since S12(1) = C · Δ and S2(11) = C · f11, the cusp forms Δ and f11 are examples of normalized eigenforms. n ∈ For a cusp form f = n anq Sk(Γ1(N)), the L-series is defined as a Dirichlet series ∞ −s L(f,s)= ann . n=1 k +1 It is known to converge absolutely on Re s> as a consequence of the Ramanujan 2 n ∈ conjecture. If f = n anq Sk(N,ε) is a normalized eigen form, the L-series L(f,s) has an Euler product −s k−1 −2s −1 −s −1 L(f,s)= (1 − app + ε(p)p p ) × (1 − app ) .

p¹N p|N

6 1.4 Galois representations To state the existence of a Galois representation associated to a modular form, we introduce some terminologies on Galois representations (cf. [12] Chapter 9). Let p be Q → Q a prime number. A choice of an embedding p deﬁnes an embedding GÉp =

Q Q → Q Q É Gal( p/ p) GÉ = Gal( / ). The Galois group G p thus regarded as a subgroup

of GÉ is called the decomposition group. It is well-deﬁned upto conjugacy. Q ⊂ Qur Q ⊂ Q The intermediate extension p p = p(ζm; p m) p deﬁnes a normal

Q Qur ⊂ É subgroup Ip = Gal( p/ p ) GÉp called the inertia subgroup. The quotient G p /Ip = Qur Q F F Gal( p / p) is canonically identified with the absolute Galois group Gp = Gal( p/ p) F ∈ p ∈ F of the residue field p. The element ϕp Gp defined by ϕ(a)=a for all a p is

called the Frobenius substitution. It is a free generator of Gp in the sense that the Z Z Z → map =− lim→ n /n Gp defined by sending 1 to ϕp is an isomorphism. Let be a prime, Eλ be a finite extension of Q and V be an Eλ-vector space → of finite dimension. We call a continuous representation GÉ GLEλ V an -adic

representation of GÉ . The group GLEλ V is isomorphic to GLn(Eλ) as a topological group if n = dimEλ V . We say that an -adic representation is unramified at a prime number p if the restriction to the inertia group Ip is trivial. In the following, we only consider -adic representations unramified at every prime p N for some integer N ≥ 1. For a prime p N, the polynomial det(1 − ϕpt : V ) ∈ Eλ[t] is well-defined. n ∈ Definition 1.2 Let f = n anq Sk(N,ε) be a normalized eigen cusp form and Q(f) → Eλ be an embedding to a finite extension of Q. A 2-dimensional -adic representation V over Eλ is said to be associated to f if, for every p N, V is unramified at p and Tr(ϕp : V )=ap(f). The goal in this note is to explain the geometric proof of the following theorem. Theorem 1.3 Let N ≥ 1 and k ≥ 2 be integers and ε :(Z/N Z)× → C× be a character. Let f ∈ Sk(N,ε) be a normalized eigenform and λ| be place of Q(f). Then, there exists an -adic representation Vf,λ over Q(f)λ associated to f. The following is a consequence of the geometric construction and the Weil conjecture. Corollary 1.4 (Ramanujan’s conjecture (see Corollary 4.5)) For every prime p, we have 11 |τ(p)|≤2p 2 . Here is a reason why Frobenius’s are so important. Theorem 1.5 (Cebotarev’s density theorem) Let L be a finite Galois extension of Q and C ⊂ Gal(L/Q) be a conjugacy class. Then there exist infinitely many prime p such that L is unramifed at p and that C is the class of ϕp.

7 If L = Q(ζN ), this is equivalent to Dirichlet’s Theorem on Primes in Arithmetic Progressions. The following is a consequence of Theorem 1.5. Let V1 and V2 be -adic represen- ≥

tations of GÉ . If there exists an integer N 1 such that

Tr(ϕp : V1)=Tr(ϕp : V2)

ss ss for every prime p N, the semi-simpliﬁcations V1 and V2 are isomorphic to each other. In particular, the -adic representation associated to f is unique upto isomorphism, since it is known to be irreducible by a theorem of Ribet [17]. It also follows that we may replace the condition Tr(ϕp : V )=ap(f) in Deﬁnition 1.2 by a stronger one k−1 2 det(1 − ϕpt : V )=1− ap(f)t + ε(p)p t .

2 Modular curves and modular forms

Modular forms are defined as certain holomorphic functions on the Poincaré upper half plane. To link them to Galois representations, we introduce modular curves. Modular curves are defined as the moduli of elliptic curves.

2.1 Elliptic curves We define elliptic curves. Basic references for elliptic curves are [22], [16] Chapter 2. First, we consider elliptic curves over a field k of characteristic =2 , 3. An elliptic curve over k is the smooth compactification of an affine smooth curve defined by

y2 = x3 + ax + b where a, b ∈ k satisfying 4a3 +27b2 = 0. Or equivalently,

2 3 y =4x − g2x − g3

∈ 3 − 2 where g2,g3 k satisfying g2 27g3 =0. 2 More precisely, E is the curve in the projective plane Pk defined by the homogeneous equation Y 2Z = X3 + aXZ2 + bZ3. The point O =(0:1:0)∈ E(k) is called the 0-section. Precisely speaking, an elliptic curve is a pair (E,O) of a projective smooth → 2 curve E of genus 1 and a k-rational point O. The embedding E Pk is defined by the 2 3 basis (x, y, 1) of Γ(E,OE(3O)). For an elliptic curve E defined by y =4x − g2x − g3, the j-invariant is defined by g3 j(E)=123 2 . 3 − 2 g2 27g3 We define an elliptic curve over an arbitrary base scheme S. An elliptic curve over S is a pair (E,O) of a proper smooth curve f : E → S of genus 1 and a section

8 → O O 1 ∗ 1 O : S E. We have f∗ E = S and f∗ΩE/S = O ΩE/S = ωE is an invertible OS-module. An elliptic curve has a commutative group structure. To define the addition, we introduce the Picard functor ([2] Chapter 8). For a scheme X, the Picard group Pic(X) is the isomorphism class group of invertible OX -modules. The addition is defined by the tensor product. If X is a smooth proper curve over a field k, the Picard group Pic(X) is equal to the divisor class group Coker(div : k(X)× → Z). x:closed points of X ∈ × · For a non-zero rational function f k(X) , its divisor divf is defined to be x ordxf → Z Z → Z [x]. The degree map deg : Pic(X) is induced by the map x:closed points of X , whose x-component is the multiplication by [κ(x):k]. Let E be an elliptic curve over a scheme S. For a scheme T over S, the degree map deg : Pic(E ×S T ) → Z(T ) has a section Z(T ) → Pic(E ×S T ) defined by 1 → [O(O)]. O L L → Z For an invertible E×ST -module , its degree deg : T is the locally constant L L| ∗ × → function defined by deg (t) = deg( E×T t). The pull-back O : Pic(E S T ) Pic(T ) ∗ also has a section f : Pic(T ) → Pic(E ×S T ). Thus, we have a decomposition

× Z ⊕ ⊕ 0 Pic(E S T )= (T ) Pic(T ) PicE/S(T )

0 → and a functor PicE/S : (Schemes/S) (Abelian groups) is deﬁned. We deﬁne a → 0 ∈ morphism of functors E PicE/S by sending P E(T ) to the projection of the class O ∈ × [ ET (P )] Pic(E S T ).

→ 0 Theorem 2.1 (Abel’s theorem) (cf. [16] Theorem 2.1.2) The morphism E PicE/S of functors is an isomorphism.

0 → L ∈ 0 The inverse PicE/S E is defined as follows. For [ ] PicE/S(T ), the support of ∗ L →L → × the cokernel of the natural map fT fT ∗( (O)) (O) defines a section T E S T . 0 → 0 Since PicE/S is a sheaf of abelian groups, the isomorphism E PicE/S defines a group structure on the scheme E over S. For a morphism f : E → E, the pull-back ∗ 0 → 0 ∗ → → map f : PicE/S PicE/S defines the dual f : E E. The map f : E E itself is identified with the push-forward map f∗ : E → E induced by the norm map recalled ∗ ∗ later in §3.2. We have f ◦ f = [deg f]E and hence f ◦ f = [deg f]E. If an elliptic curve E over a field k is defined by y2 = x3 + ax + b, the addition on E(k) is described as follows. Let P, Q ∈ E(k). The line PQmeets E at the third point R. The divisor [P ]+[Q]+[R] is linearly equivalent to the divisor [O]+[R]+[R], where R is the opposite of R with respect to the x-axis. Thus, we have [P ]+[Q]+[R]= [O]+[R]+[R] in Pic(E) and ([P ] − [O]) + ([Q] − [O]) = [R] − [O] in Pic0(E). Hence we have P + Q = R in E(k). We introduce the Weil pairing (cf. [16] Section 2.8). For an elliptic curve f : E → S and an integer N ≥ 1, the Weil pairing is a non-degenerate alternating pairing

9 ( , )E[N] : E[N] × E[N] → μN . Here and in the following, E[N] and μN denote the kernels of the multiplications [N]:E → E and [N]:Gm → Gm. Non-degerate means ∗ that the induced map E[N] → E[N] = Hom(E[N],μN ) to the Cartier dual (see [4] Chapter V Section (3.8)) is an isomorphism of finite group schemes. Let P ∈ E[N](S)beanN-torsion point and L be an invertible OE-module corre- ∗ ∗ sponding to P . Since [[N] L]=NP = 0, the invertible OE-module [N] L is canonically ∗ isomorphic to the pull-back of an invertible OS -module f∗[N] L. For another N-torsion point Q ∈ E[N](S), the translation Q+ satisfies [N]◦(Q+) = [N]. Hence it induces an ∗ ∗ ∗ automorphisms Q of [N] L and of f∗[N] L by pull-back. Thus we obtain a morphism ∗ ∗ (P, )E[N] : E[N] → μN ⊂ Gm = Aut f∗[N] L sending Q to Q , that defines a bilinear pairing ( , )E[N] : E[N] × E[N] → μN .

2.2 Elliptic curves over C To give an elliptic curve over C is equivalent to give a complex torus of dimension 1, as follows. For more detail, see [22] Chapter VI and [12] Section 1.4. Let E be an elliptic curve over C. Then, E(C) is a connected compact abelian complex Lie group of dimension 1. The tangent space Lie E of E(C) at the origin is a C- vector space of dimension 1. The exponential map exp : Lie E → E(C) is surjective and the kernel is a lattice of E(C) and is identified with the singular homology H1(E(C), Z). A lattice L of a complex vector space V of finite dimension is a free abelian subgroup generated by an R-basis. Conversely, let L be a lattice of C. The ℘-function is defined by 1 1 1 x = ℘(z)= + − . z2 (z − ω)2 ω2 ω∈L,ω=0 Since d℘(z) 1 y = = −2 , dz (z − ω)3 ω∈L they satisfy the Weierstrass equation

2 3 y =4x − g2x − g3 1 1 where g =60 and g = 140 .IfL = Z + Zτ for τ ∈ H, we have 2 ω4 3 ω6 ω∈L,ω=0 ω∈L,ω=0

(2πi)4 1 (2πi)4 g2 =60G4(τ)=60· E4 = E4, 3! 120 12 (2πi)6 1 (2πi)6 g = 140G (τ) = 140 · − E = − E 3 6 5! 252 6 63 6 and hence 1 g3 − 27g2 =(2πi)12 (E3 − E2)=(2πi)12Δ =0 . 2 3 123 4 6 10 2 3 Thus the equation y =4x − g2x − g3 deﬁnes an elliptic curve E over C. The map C/L → E(C) deﬁned by z → (℘(z),℘(z)) is an isomorphism of compact Riemann surfaces. ∈ C 1 We show that the Weil pairing (P,√ Q) μN ( ) for the N-torsion points P = N ,Q= τ ∈ C 2π −1 N E = / 1,τ is equal to exp N . The elliptic function

σ(z) σ(z − 1 (i + jτ + 1 )) f(z)= N N σ(z − 1) σ(z − 1 (i + jτ)) i,j=1,... ,N N

∗ is a basis of [N] L for L = OE(P −O) where σ denotes the Weierstrass’s σ-function for the lattice 1,τ (for the deﬁnition, see [22] p. 156). Hence, the Weil pairing (P, Q)is f(z + τ ) η(1)τ − η(τ)1 equal to the ratio N . We see that it is equal to exp , by using the f(z) N σ(z + ω)σ(w) formula = exp(η(ω)(z−w)) for z,w ∈ C and ω ∈1,τ ([23] Proposition σ(z)σ(w + ω) 5.4 (b)). where η denotes the Dedekind η-function (for the deﬁnition, see loc. cit.√ p. 65). Thus the assertion follows from the Legendre relation η(1)τ − η(τ)1 = 2π −1 (loc. cit. Proposition 5.2 (d)).

2.3 Modular curves over C The set of isomorphism classes of elliptic curves over C has a one-to-one correspondence with the quotient of the Poincaré upper half plane by SL2(Z). The j-invariant defines an isomorphism SL2(Z)\H → C of Riemann surfaces. Modular curves over C are defined as finite coverings of an algebraic curve SL2(Z)\H over C. More detail is found in [12] Chapters 2 and 3. We put

×2 ω1 R = {lattices in C}, R = {(ω1,ω2) ∈ C |Im > 0}. ω2

The multiplication deﬁnes an action of C× on R and on R. We consider the map R→R sending (ω1,ω2) to the latticeω1,ω2 generated byω1,ω2. A natural action ab ω1 aω1 + bω2 of SL2(Z)onR is deﬁned by = . It induces a bijection cd ω2 cω1 + dω2 SL2(Z)\R→R. The map H → R : τ → (τ,1) is compatible with the action of SL2(Z) and induces bijections

× × × H → C \R,SL2(Z)\H → (SL2(Z) × C )\R→C \R.

The map sending a lattice L to the isomorphism class of the elliptic curve C/L deﬁnes bijections

× SL2(Z)\H → C \R → {isomorphism classes of elliptic curves over C}.

11 The quotient Y (1)(C)=SL2(Z)\H is called the modular curve of level 1. The map

j : SL2(Z)\H → C deﬁned by the j-invariant g (τ)3 E3 j τ 2 4 ( ) = 1728 3 2 = g2(τ) − 27g3(τ) Δ is an isomorphism of Riemann surfaces ([19] Chapter 7 Proposition 5). For an integer N ≥ 1, similarly the map sending (ω1,ω2) ∈ R to the pair (E,P)= ω C/ω ,ω , 2 deﬁnes a bijection 1 2 N \ → × C× \R Γ1(N) H (Γ1(N) ) isom. classes of pairs (E,P) of an elliptic curve → . E over C and a point P ∈ E(C) of order N cω + dω ω ab Note that 1 2 ≡ 2 mod ω ,ω since c ≡ 0,d≡ 1modN for ∈ Γ (N). N N 1 2 cd 1 The quotient Γ1(N)\H is denoted by Y1(N)(C) and is called the modular curve of level Γ1(N). × The diamond operators act on Y1(N)(C). For d ∈ (Z/N Z) , the action of d × is given by d(E,P)=(E,dP). The quotient Γ0(N)\H =(Z/N Z) \Y1(N)(C)is denoted by Y0(N)(C) and is called the modular curve of level Γ0(N). We have a natural bijection isom. classes of pairs (E,C) of an elliptic curve E Γ (N)\H → . 0 over C and a cyclic subgroup C ⊂ E(C) of order N

1 We have finite flat maps Y1(N) → Y0(N) → Y (1) = A of open Riemann surfaces. The degrees of the maps are given by × ϕ(N)/2ifN ≥ 3 [Y1(N):Y0(N)] = (Z/N Z) /{±1} = 1ifN ≤ 2, and [Y0(N):Y (1)] = [SL2(Z):Γ0(N)]. Let X1(N) and X0(N) be the compactifications of Y1(N) and Y0(N). The maps 1 Y1(N) → Y0(N) → Y (1) = A are uniquely extended to finite flat maps X1(N) → 1 X0(N) → X(1) = P of compact Riemann surfaces, or equivalently of projective smooth curves over C. 1 1 We identify f ∈ S2(N) with f · 2πidτ ∈ Γ(X0(N), Ω ) and S2(N)=Γ(X0(N), Ω ). 1 Applying the Riemann-Hurwitz formula to the map j : X0(N) → X(1) = P ,we obtain the genus formula 1 1 1 1 g(X (N)) = g (N)=1+ [SL (Z):Γ(N)] − ϕ∞(N) − ϕ (N) − ϕ (N) 0 0 12 2 0 2 3 6 4 4 12 where 0if9|N or if ∃p|N,p ≡−1mod3 ϕ6(N)= {p|N : p ≡ 1mod3} 2 if otherwise, 0if4|N or if ∃p|N,p ≡−1mod4 ϕ4(N)= {p|N : p ≡ 1mod4} 2 if otherwise. and ϕ∞(NM)=ϕ∞(N)ϕ∞(M)if(N,M) = 1 and, for a prime p and e>0, (e−1)/2 e 2p if e odd ϕ∞(p )= (p +1)pe/2−1 if e even.

We have g0(11) = 1 and hence X0(11) is an elliptic curve, defined by the equation 2 3 − 124 − 2501 124 3 − 2501 2 − 5 y =4x 3 x 27 , where Δ = 3 27 27 = 11 . The space S2(11) = 1 Γ(X0(11), Ω ) of cusp forms of weight 2 and level 11 is generated by the differential dx form corresponding to f . y 11 The universal elliptic curves over the modular curves Y1(N) for N ≥ 4 are defined 2 as follows. We consider the semi-direct product Γ1(N) Z with respect to the left t −1 × 2 action by γ . We define an action of C × Γ1(N) Z on R×C by

c((ω1,ω2),z)=((cω1,cω2),cz)

γ((ω1,ω2),z)=((aω1 + bω2,cω1 + dω2),z)

(m, n)((ω1,ω2),z)=((ω1,ω2),z+ mω1 + nω2). ab for c ∈ C×, γ = ∈ Γ (N) and (m, n) ∈ Z2. The projection R× C → R is cd 1 × 2 × compatible with the projection C × Γ1(N) Z → C × Γ1(N). Assume N ≥ 4. By taking the quotient, we obtain

Z2 \ × C → \ EY1(N) =(Γ1(N) ) (H ) Y1(N)=Γ1(N) H.

The fiber at τ ∈ H is the elliptic curve C/Z+Zτ.IfN =1, 2, we have −1 ∈ Γ1(N) and the general fiber is the quotient of C/Z + Zτ by the involution −1 and is isomorphic to P1.ForN = 3, the fibers at primitive cubic roots τ = ω, ω−1 are also isomorphic to P1.

The universal elliptic curve EY1(N) has the following modular interpretation.

Lemma 2.2 Assume N ≥ 4.Letf : E → S be a holomorphic family of elliptic curve and P : S → E be a section of exact order N. Then, there exists a unique morphism S → Y1(N) such that (E,P) is isomorphic to the pull-back of the universal elliptic ω curve E and the section deﬁned by z = 2 . Y1(N) N

13 ∗ 1 Sketch of Proof. Since the question is local on S, we may take basis of ω = O ΩE/S 1 and of R f∗Z. Then, they give a family of lattices in C parametrized by S and define × a map S → R. The induced map S → Y1(N)=(Γ1(N) × C )\R is well-defined and satisfies the condition.

2.4 Modular curves and modular forms The deﬁnition of modular forms given in Section 1.1 is rephrased in terms of the universal elliptic curves as follows. For more detail, we refer to [6] Chapitre VII, Sections 3 and 4 and also [11] Sections 12.1, 12.3. ≥ ∗ → Let N 4. Let ωY1(N) be the invertible sheaf O ΩEY1(N)/Y1(N) where O : Y1(N) EY1(N) is the 0-section of the universal elliptic curve. Then, we have

⊗k {f : H → C|f is holomorphic and satisfies (1) in Definition 1.1} =Γ(Y1(N),ω ) √ · − ⊗k ⊗2 → by identifying√ f with√f (2π 1dz) . By the isomorphism ω ΩY1(N) sending ⊗2 ⊗k−2 (2π −1dz) → 2π −1dτ, the left hand side is identified with Γ(Y1(N),ω ⊗

ΩY1(N)). ≥ → Assume N 5. Then the universal elliptic curve EY1(N) Y1(N) is uniquely → extended to a smooth group scheme EY1(N) X1(N) whose ﬁbers at cusps are Gm. √ dq ∗ − ⊗2 → Let ωX1(N) = O Ω . Since 2π 1dτ = , the isomorphism ω ΩY1(N) EY1(N)/X1(N) q ⊗2 → on Y1(N) is extended to an isomorphism ωX1(N) ΩX1(N)(log(cusps)) on X1(N). By ⊗2 → the isomorphism ωX1(N) ΩX1(N)(log(cusps)), we may identify

⊗k ⊃ ⊗k−2 ⊗ Mk(Γ1(N))=Γ(X1(N),ω ) Sk(Γ1(N))=Γ(X1(N),ω ΩX1(N)).

Since X1(N) is compact, the C-vector spaces Mk(Γ1(N)) and Sk(Γ1(N)) are of ﬁnite dimension. For N ≥ 5, there exists a constant C satisfying deg ω = C · [SL2(Z):Γ1(N)]. The ⊗2 → 1 isomorphism ω ΩX1(N)(log cusps) implies 1 N 2g (N) − 2+ ϕ( )ϕ(d)=2C · [SL (Z):Γ(N)]. 1 2 d 2 1 d|N

In particular, for p ≥ 5, we have

2 2g1(p) − 2+p − 1=2C · (p − 1).

1 Since g1(5) = 0, we have C = 24 and ⎧ ⎪ 1 1 N ⎨1+ [SL (Z):Γ(N)] − ϕ( )ϕ(d)ifN ≥ 5, 24 2 1 4 d dim S (Γ (N)) = g (N)= | 2 1 1 ⎩⎪ d N 0ifN ≤ 4.

14 By Riemann-Roch, we have

⊗(k−2) 1 dim Sk(Γ1(N)) = deg(ω ⊗ Ω )+χ(X1(N), O)=(k − 2) deg ω + g1(N) − 1 k − 1 1 N = [SL (Z):Γ(N)] − ϕ( )ϕ(d) 24 2 1 4 d d|N for k ≥ 3,N ≥ 5.

1 2.5 Modular curves over Z[N ] To construct Galois representations associated to modular forms, we descend the defi- Q Z 1 nition field of modular curves to and consider their integral models over [ N ]. ≥ Z 1 → Let N 1 be an integer and T be a scheme over [ N ]. We say a section P : T E of an elliptic curve E → T is exactly of order N,ifNP = 0 and if Pt ∈ Et(t)isof ∈ M Z 1 → order N for every point t T . We define a functor 1(N) : (Scheme/ [ N ]) (Sets) by isomorphism classes of pairs (E,P) of an elliptic curve M (N)(T )= . 1 E → T and a section P ∈ E(T ) exactly of order N

Theorem 2.3 ([16] Corollaries 2.7.3 and 4.7.1) For an integer N ≥ 4, the functor M Z 1 1(N) is representable by a smooth aﬃne curve over [ N ].

1 Namely, there exist a smooth aﬃne curve Y (N) 1 over Z[ ] and a pair (E,P) 1 [N ] N

of elliptic curves E → Y (N) 1 and a section P : Y (N) 1 → E exactly of order N 1 [N ] 1 [N ] such that the map

Hom 1 (T,Y (N) 1 ) →M(N)(T ) Scheme/[N ] 1 [N ] 1

∗ ∗ sending f : T → Y (N) 1 to the class of (f E,f P ) is a bijection for every scheme T 1 [N ] Z 1 over [ N ]. If N ≤ 3, the functor M1(N) is not representable because there exists a pair (E,P) ∈M1(N)(T ) with a non-trivial automorphism. More precisely, by étale descent, there exist 2 distinct elements (E,P), (E ,P ) ∈M1(N)(T ) whose pull-backs are the same for some étale covering T → T . Proof of Theorem for N = 4. Let E → T be an elliptic curve over a scheme T Z 1 over [ 2 ] and P be a section of exact order 4. We take a coordinate of E so that 2P =(0, 0),P =(1, 1), 3P =(1, −1) and let dy2 = x3 + ax2 + bx + c be the equation defining E. Then the line y = x meets E at 2P and is tangent to E at P . Thus we have x3 +(a − d)x2 + bx + c = x(x − 1)2. Namely, E is defined by dy2 = x3 +(d − 2)x2 + x. 1 1 The moduli Y (4) 1 is given by SpecZ[ ][d, ].

1 [4 ] 4 d(d − 4)

15 To prove the general case, we consider the following variant. For an elliptic curve E and an integer r ≥ 1, let E[r] = Ker([r]:E → E) denote the kernel of the multiplication by r. We define a functor M(r) : (Scheme/Z[ 1 ]) → (Sets) by r isom. classes of pairs (E,(P, Q)) of an elliptic curve E → T M(r)(T )= . and P, Q ∈ E(T ) defining an isomorphism (Z/rZ)2 → E[r] Theorem 2.4 For an integer r ≥ 3, the functor M(r) is representable by a smooth 1 affine curve Y (r) 1 over Z[ ]. [r ] r Sketch of Proof. If r = 3, the functor M(r) is representable by the smooth affine Z 1 1 Z 1 ⊂ 2 curve Y (3) = Spec [ 3 ,ζ3][μ, μ3−1 ] over Spec [ 3 ]. The universal elliptic curve E P is defined by X3 + Y 3 + Z3 − 3μXY Z and the origin is O =(0, 1, −1). The basis of E[3] is given by P =(0, 1, −ω2) and Q =(1, 0, −1). Next we consider the case r = 4. Let E be the universal elliptic curve over Y1(4) and P be the universal section of order 4. Then, the Weil pairing (P, ) defines a map E[4] → μ4. The modular curve Y (4) is the inverse image of the complement of the open and closed subscheme μ2 ⊂ μ4. If r is divisible by s = 3 or 4, one can construct Y (r) 1 as a finite étale scheme [r ]

over Y (s) 1 . For general r ≥ 5, the modular curve Y (r) 1 is obtained by patching [r ] [r ]

the quotient Y (r) 1 = Y (sr) 1 /Ker(GL (Z/rsZ) → GL (Z/rZ)) for s =3, 4. [sr ] [sr ] 2 2 1 By the Weil pairing, the scheme Y (r) 1 is naturally a scheme over Z[ ,ζ ]. For [r ] r r r =1, 2, the modular curves Y (r) 1 are also deﬁned by patching the quotients. The j- [r ] → A1 2 − −

invariant defines an isomorphism Y (1) . The Legendre curve y = x(x 1)(x λ) 1 1 defines an isomorphism SpecZ[ ][λ, ] → Y (2) 1 . 2 λ(λ−1) [2 ] By regarding P, Q ∈ E[N] as a map (Z/N Z)2 → E[N], we define a natural right action of GL2(Z/N Z)onM(N) and hence on Y (N) as that induced by the natural 2 action of GL2(Z/N Z)on(Z/N Z) . The modular curve Y (N) 1 is constructed as the quotient 1 [ ] N ab Y (N) 1 ∈ GL (Z/N Z) a =1,c=0 . 2 [N ] cd

The modular curve Y (N) 1 for N ≤ 3 are also deﬁned as the quotients. 1 [N ]

The Atkin-Lehner involution w : Y (N) 1 → Y (N) 1 is defined by send- N 1 [N ,ζN ] 1 [N ,ζN ] ing (E,P)to(E/P ,Q) where Q ∈ E[N]/P ⊂(E/P )[N] is the inverse image of ζN ∈ μN by the isomorphism (P, ):E[N]/P →μN . The affine smooth curve Y (N) 1 is uniquely embedded in a proper smooth curve 1 [N ] 1 X (N) 1 over Z[ ] ([6] Chapitre IV 4.14, [11] Section 9). The universal ellip- 1 [N ] N 1 tic curve E 1 1 over Y (N) is uniquely extended to a smooth group scheme Y (N) [ ] 1 [ ] N N E over X (N) 1 such that the fiber at every geometric point in the com-

X1(N) [ 1 1 [ ] ] N N

plement X (N) 1 \ Y (N) 1 is isomorphic to G (loc. cit.). We deﬁne an invert- 1 [N ] 1 [N ] m ∗ 1 Q ible sheaf ωX1(N) [ 1 ] to be the inverse image O ΩEX (N) /X1(N) 1 . The -vector [ ] N 1 [ 1 ] N N

⊗k−2 ⊗ 1 Q C É space Sk(Γ1(N))É =Γ(X1(N) ,ω Ω ) gives a -structure of the -vector space

⊗k−2 ⊗ 1 Sk(Γ1(N)) =Γ(X1(N) ,ω Ω ).

2.6 Hecke operators In this subsection, we give an algebraic deﬁnition of the Hecke operators. For more detail, we refer to [12] Section 7.9. 1 For integers N,n ≥ 1, we deﬁne a functor T (N,n) 1 : (Schemes/Z[ ]) → (Sets) 1 [N ] N by

T (N,n) 1 (T )

1 ⎧ [N ] ⎫ ⎨isom. classes of triples (E,P,C) of an elliptic curve E over T ,a⎬ → = ⎩ section P : T E exactly of order N and a subgroup scheme ⎭ C ⊂ E ﬁnite ﬂat of degree n over T such that P ∩C = O

and a morphism s : T (N,n) 1 →M(N) 1 of functors sending (E,P,C)to(E,P). 1 [N ] 1 [N ]

The functor T (N,n) 1 is representable by a ﬁnite ﬂat scheme T (N,n) 1 over 1 [N ] 1 [N ]

Y (N) 1 ,ifN ≥ 4. The map T (N,n) 1 → Y (N) 1 is uniquely extended to

1 [N ] 1 [N ] 1 [N ]

a finite flat map of proper normal curves s : T (N,n) 1 → X (N) 1 . 1 [N ] 1 [N ] For an elliptic curve E → T and a subgroup scheme C ⊂ E finite flat of degree n, the quotient E = E/C is defined and the induced map E → E is finite flat of ∗ ∗ O − O →O degree n. The structure sheaf E is the kernel of pr1 μ : E E×T C where × → pr1,μ : E T C E denote the projection and the addition respectively. By this construction, we may identify the set T (N,n) 1 (T ) with 1 [N ] ⎧ ⎫ ⎨ isom. classes of pairs (ϕ : E → E,P) of finite flat morphism ⎬ → ⎩ ϕ : E E of elliptic curves over T of degree n and a section ⎭ → ∩ → P : T E exactly of order N such that P Ker(E E )=O .

We define a morphism t : T (N,n) 1 →M(N) 1 of functors sending (ϕ : E → 1 [N ] 1 [N ] E ,P)to(E ,ϕ(P )), It also induces a finite flat map of proper curves t : T (N,n) 1 → 1 [N ] X (N) 1 . 1 [N ] For an integer n ≥ 1, we define the Hecke operator Tn : Sk(Γ1(N)) → Sk(Γ1(N)) ∗ ∗ as s∗ ◦ ϕ ◦ t . Here

∗ ⊗k−2 1 ∗ ⊗k−2 1 t : Sk(Γ1(N))=Γ(X1(N),ω ⊗ Ω ) → Γ(T 1(N,n),t ω ⊗ Ω ),

∗ ⊗k−2 1 ⊗k−2 1 s∗ :Γ(T 1(N,n),s ω ⊗ Ω ) → Γ(X1(N),ω ⊗ Ω )=Sk(Γ1(N))

are induced by the maps s, t : T (N,n) 1 → X (N) 1 deﬁned above respectively. 1 [ ] 1 [ ] N N ∗ ∗ The push-forward map s∗ is induced by the trace map. Since s ω = ωE and t ω = ωE , the map ϕ : E → E induces

∗ ∗ ⊗k−2 1 ∗ ⊗k−2 1 ϕ :Γ(T 1(N,n),t ω ⊗ Ω ) → Γ(T 1(N,n),s ω ⊗ Ω ).

17 × The group (Z/N Z) has a natural action on the functor M1(N). Hence it acts on × Sk(Γ1(N)). For d ∈ (Z/N Z) , the action is denoted by d and called the diamond operator. § ∈ 1 ∈ We verify the equality (1) in 1.3 for Tp.Forτ H, the point P = N E = C C τ+i / 1,τ is of order N. The elliptic curve E = / 1,τ has p + 1 subgroups p ,i= − 1 0,... ,p 1 and p of order p.Ifp N, each of them defines a point of T1(N,p) and ∈ | 1 they are the inverse image of τ H by s.Ifp N, the subgroup p is a subgroup 1 of N and does not define a point of T1(N,p). The other p subgroups define points of T1(N,p) and they are the inverse image of τ ∈ H by s.Fori =0,... ,p− 1, the τ+i → C τ+i 1 subgroups p define E / 1, p and the image of P is N . Thus their images τ+i 1 → C 1 by t are p .Ifp N, the subgroup p define E / p ,τ . The multiplication by C 1 →C p p induces an isomorphism / p ,τ / 1,pτ and the image of P is N . Thus its image by t is ppτ. From this the equality (1) follows immediately. We define the Hecke algebra by × Tk(Γ1(N)) = Q[Tn,n∈ N, d,d∈ (Z/N Z) ] ⊂ EndSk(Γ1(N)). It is a finite commutative Q-algebra. Proposition 2.5 The map

→ C É Sk(Γ1(N)) Hom (Tk(Γ1(N)), ) (2) sending a cusp form f to the linear form T → a1(Tf) is an isomorphism of Q-vector spaces.

× → C Proof. Suﬃces to show that the pairing Sk(Γ1(N)) Tk(Γ1(N)) deﬁned → ∈ by (T,f) a1(Tf) is non-degenerate. If f Sk(Γ1(N)) is in the kernel, an(f)= n ∈ a1(Tnf) = 0 for all n and f = n an(f)q =0.IfT Tk(Γ1(N)) is in the kernel, ∈

Tf is in the kernel for all f Sk(Γ1(N)) since a1(T Tf)=a1(TT f) = 0 for all T ∈ Tk(Γ1(N)). Hence Tf = 0 and T =0. Corollary 2.6 The isomorphism (2) induces a bijection of ﬁnite sets

{ ∈ | }→ C É f Sk(Γ1(N)) normalized eigenform Hom -algebra (Tk(Γ1(N)), ) (3) Proof. We prove the bijectivity. Let ϕ be the linear form corresponding to f. The condition ϕ(1) = 1 is equivalent to a1(f) = 1. If ϕ is a ring homomorphism, we have an(Tf)=a1(TnTf)=ϕ(TnT )=ϕ(T )ϕ(Tn)=ϕ(T )a1(Tnf)=ϕ(T )an(f) for every ≥ ∈ n n n 1 and for T Tk(Γ1(N)). Thus, we have Tf = n an(Tf)q = n ϕ(T )an(f)q = ϕ(T )f and f is a normalized eigenform. Conversely, if f is a normalized eigenform and Tf = λT f for each T ∈ Tk(Γ1(N)), we have ϕ(T )=a1(Tf)=a1(λT f)=λT a1(f)= λT .Thusϕ is a ring homomorphism. Since Tk(Γ1(N)) is ﬁnite over Q, the right hand side is a ﬁnite set. ∈ Q ⊂ C

For a normalized eigenform f Sk(Γ1(N)) , the subﬁeld (f) is the image of the corresponding Q-algebra homomorphism ϕf : Tk(Γ1(N)) → C and hence is a ﬁnite extension of Q.

18 3 Construction of Galois representations: the case k =2

We will construct Galois representations associated to modular forms in the case k =2, using the Tate module of the Jacobian of a modular curve.

3.1 Galois representations and finite étale group schemes To construct Galois representations, it suffices to define group schemes, in the following sense. For a field K, we have an equivalence of categories

(finite étale commutative group schemes over K) → (finite GK-modules) → → defined by A A(K). The inverse is given by M Spec(HomGK (M,K)). In the case K = Q, it induces an equivalence 1 finite GÉ -modules (finite étale commutative group schemes over Z[ ]) → N unramified at p N for N ≥ 1.

Lemma 3.1 Let N ≥ 1 be an integer and A be a ﬁnite ´etale group scheme over Z 1 Q F Spec [ N ]. For a prime number p N, the action of ϕp on A( )=A( p) is the same

→ as that deﬁned by the geometric Frobenius endomorphism Fr : Ap A p .

Proof. Clear from the commutative diagram O −−−→ F ⏐A p ⏐p ⏐ ⏐ a →ap x →xp O −−−→ F . A p p

To define an -adic representation of GÉ unramified at p N, it suffices to construct Z nZ Z 1 an inverse system of finite étale / -module group schemes over [ N ].

3.2 Jacobian of a curve and its Tate module

First, we consider the case g0(N) = 1, e.g. N = 11. Then, E = X0(N) is an elliptic Q ⊗ n Q curve and the Tate module VE = ←lim− nE[ ]( ) deﬁnes a 2-dimensional -adic representation. To construct the Galois representation for general N, we will introduce the Jacobian.

19 Let f : X → S be a proper smooth curve with geometrically connected ﬁbers of genus g. For simplicity, we assume f : X → S admits a section s : S → X. We deﬁne 0 → a functor PicX/S : (Schemes/S) (Abelian groups) by × → Z 0 Ker(deg : Pic(X S T ) (T )) PicX/S (T )= ∗ . Im(f : Pic(T ) → Pic(X ×S T )) A section s : S → X provides a decomposition × Z ⊕ ⊕ 0 Pic(X S T )= (T ) Pic(T ) PicX/S (T ) depending on the choice of a section.

0 Theorem 3.2 ([2] Proposition 9.4/4) The functor PicX/S is representable by a proper smooth scheme J = JacX/S with geometrically connected ﬁbers of dimension g.

The proper group scheme (=abelian scheme) JacX/S is called the Jacobian of X.If g = 1, Abel’s theorem says that the canonical map E → JacE/S is an isomorphism. Let f : X → Y be a finite flat morphism of proper smooth curves. The pull- ∗ back of invertible sheaves defines the pull-back map f : JacY/S → JacX/S . We also have a push-forward map defined as follows. The norm map f∗ : f∗Gm,X → Gm,Y defines a push-forward of Gm-torsors and a map Pic(X) → Pic(Y ), for a finite flat map f : X → Y of schemes (see [10] 7.1). They define a map of functors and hence ∗ a morphism f∗ : JacX/S → JacY/S (see loc. cit. 7.2). The composition f∗ ◦ f is the multiplication by deg f. If f : X → Y is a finite flat map of proper smooth curves over a field, then the × → Z → isomorphism Coker(div : k(X) x ) Pic(X) has the following compatibility. ∗ ∗ × The pull-back f : Pic(Y ) → Pic( X) is compatible with the inclusion f : k(Y ) → × Z → Z · k(X) and the map y x sending the class [y]to x →y e(x/y) [x] where e(x/y) denotes the ramification index. The push-forward f∗ : Pic(X) → Pic(Y )is × → × Z → Z compatible with the norm map f∗ : k(X) k(Y ) and the map x y sending the class [x]to[κ(x):κ(y)] · [y] where y = f(x). We define the Weil pairing. Let N ≥ 1 be an integer invertible on S. Then, a non- degenerate pairing JX/S [N] × JX/S [N] → μN of finite étale groups schemes is defined as follows. First, we recall that, for invertible OX -modules L and M, the pairing L, M is defined as an invertible OS-module. It is characterized by the bilinearity, the compatibility with base change and by the property L, M = fD∗(L|D)ifM = OX (D) ⊂ L ∗L L M L⊗ deg M for a divisor D X finite flat over S.If = f 0, we have , = 0 . L ∈ 0 L⊗N ∗L L ∈ If N[ ]=0 PicX/S (S), we have = f 0 for some 0 Pic(S). Hence, for M∈Pic(X) of degree 0, we have a trivialization L, M⊗N = L⊗N , M = ∗L M ∗L⊗ deg M O M ∈ 0 f 0, = f 0 = S.IfN[ ]=0 Pic (X/S), we have another trivializa- ⊗N tion L, M = OS. By comparing them, we obtain an invertible function L, MN on S, whose N-th power turns out to be 1. Thus the Weil pairing L, MN ∈ Γ(S, μN ) is defined. In the case X = E is an elliptic curve, this is the same as the Weil pairing defined before.

20 For a proper smooth curve over C, the Jacobian has an analytic description as a compact complex torus (cf.[12] Sections 6.1, 6.2). Let X be a smooth proper curve over C, or equivalently a compact Riemann surface. The canonical map

H1(X, Z) → Hom(Γ(X, Ω), C) → is deﬁned by sending a 1-cycle γ to the linear form ω γ ω. It is injective and the image is a lattice. A canonical map

0 Pic (X)=JX (C) → Hom(Γ(X, Ω), C)/Image H1(X, Z) (4) − → P is deﬁned by sending [P ] [Q] to the class of the linear form ω Q ω. This is an

isomorphism of compact complex tori. Thus, in this case, the N-torsion part JacX/ [N] of the Jacobian is canonically identified with H1(X, Z) ⊗ Z/N Z. For a finite flat map f : X → Y of curves, the isomorphism (4) has the following functoriality. The pull-back f ∗ : Pic0(Y ) → Pic0(X) is compatible with the dual of the push-forward map f∗ :Γ(X, Ω) → Γ(Y,Ω) and the pull-back map H1(Y,Z) → 0 0 H1(X, Z). The push-forward f∗ : Pic (X) → Pic (Y ) is compatible with the dual of ∗ the pull-back map f :Γ(Y,Ω) → Γ(X, Ω) and the push-forward map H1(X, Z) → H1(Y,Z). → Z ⊗ Z Z

The isomorphism JacX/ [N] H1(X, ) /N is compatible with the pull-back → and the push-forward for a ﬁnite ﬂat morphism. By the isomorphism JacX/ [N]

Z ⊗Z Z × → H1(X, ) /N , the Weil pairing JacX/ [N] JacX/ [N] μN is identified with the pairing induced by the cap-product H1(X, Z) × H1(X, Z) → Z. Now, we introduce the Tate module of the Jacobian a curve. Let X be a proper smooth curve over a field k with geometrically connected fiber of genus g. For a prime number invertible in k, we define the -adic Tate module by

n ¯ n T JacX/k =← lim− nJacX/k[ ](k) =← lim− nPic(Xk¯)[ ] and V JacX/k = Q ⊗ T JacX/k.

≥ Z 1 Corollary 3.3 Let N 1 be an integer and X be a proper smooth curve over [ N ] with

geometrically connected ﬁbers of genus g. Then, V Jac É is an -adic representation

XÉ/

of GÉ of degree 2g unramiﬁed at p N.

Proof. The multiplication [ ] : Jac 1 → Jac 1 is ﬁnite ´etale. Hence X/ [N] X/ [N]

n Q n C Z ⊗ Z nZ Z nZ 2g É JacX/ É [ ]( ) = JacX/ [ ]( )=H1(X, ) / is isomorphic to ( / ) as a Z nZ Z ⊗ Q Q2g Q

/ -module and V Jac É is isomorphic to H (X, ) as a -vector XÉ/ 1 n 1 space. Since Jac 1 [ ] is a ﬁnite ´etale scheme over Z[ ], the -adic representation X/ [N] N

V Jac É is unramiﬁed at p N.

XÉ/ In the rest of the subsection, we will see that the zeta function of a curve is expressed by the Tate module of the Jacobian. Let f : X → X be an endomorphism of a proper smooth curve over a ﬁeld k. Let Γf , Δ ⊂ X × X be the graphs of f and of the identity

21 ∈ Z and let (Γf , ΔX )X×kX be the intersection number. Then, for a prime number invertible in k, the Lefschetz trace formula (cf. §4.1) gives us − (Γf , ΔX )X×kX =1 Tr(f∗ : TJX ) + deg f.

n Assume k = Fp and apply the Lefschetz trace formula to the iterates F of the Frobenius endmorphism F : X → X. Then we obtain

n n Card X(Fpn )=1− Tr(F∗ : TJX )+p . Thus the zeta function deﬁned by ∞ CardX(F n ) Z(X, t) = exp p tn n n=1 satisﬁes det(1 − F∗t : T J ) Z(X, t)= X . (1 − t)(1 − pt) Z 1 Thus, for a proper smooth curve X over [ N ] and a prime p N, we have

det(1 − ϕ t : T J )=Z(X ⊗ 1 F ,t)(1 − t)(1 − pt). p X [N ] p

Theorem 3.4 (Weil) Let α be an eigenvalue of ϕp on √TJX . Then, α is an algebraic integer and its conjugates have complex absolute values p.

3.3 Construction of Galois representations Now we construct a Galois representation associated to a modular form. For more detail, we refer to [12] Chapter 9. We recall the Eichler-Shimura isomorphism. Proposition 3.5 The canonical isomorphism Z ⊗ R → C C

H1(X1(N), ) Hom(S2(Γ1(N)), ) = Hom(Γ(X1(N), Ω), )

is an isomorphism of T2(Γ1(N))Ê -modules. ∗ Proof. The T2(Γ1(N))-module structure is deﬁned by T on S2(Γ1(N)) and is Q ∈ deﬁned by T∗ on H1(X1(N), ) for T T2(Γ1(N)). Thus, it follows from the equality ω = f ∗ω. f∗γ γ Z Z ≥

Deﬁne the integral Hecke algebra T2(Γ1(N)) to be the -subalgebra [Tn,n ∈ Z Z × ⊂ Z Z

1; d ,d ( /N ) ] End H1(X1(N), ). The -algebra T2(Γ1(N)) is commutative and finite flat over Z. It follows from Proposition 3.5 that the Fourier coefficients an(f) → C lie in the image of the ring homomorphism ϕf : T2(Γ1(N)) for a normalized eigenform f. Hence they are algebraic integers in the number field Q(f). ≥

For an integer N 1, let J1(N) denote the Jacobian JacX1(N)/É of the modular curve X1(N) over Q.

22 Corollary 3.6 (cf. [12] Lemma 9.5.3) The T2(Γ1(N))É -module V(J1(N)) is free of rank 2.

Proof. By Propositions 2.5 and 3.5 and by fpqc descent, H1(X1(N), Q) is a free Q ⊗ Q

T2(Γ1(N))É -module of rank 2. Hence V(J1(N)) = H1(X1(N), ) is also free of rank 2. For a place λ| of Q(f), we put

V = V (J (N)) ⊗ 2 1 Q(f) . f,λ 1 T (Γ (N))É λ

The Q(f)λ-vector space Vf,λ is a 2-dimensional -adic representation unramiﬁed at p N.

Theorem 3.7 (cf. [12] Theorem 9.5.4) The -adic representation Vf,λ is associated to f. Namely, for p N, we have

2 det(1 − ϕpt : Vf,λ)=1− ap(f)t + εf (p)pt .

Proof will be given in the next subsection.

− 2 − − Corollary 3.8 If we put√ 1 ap(f)t + εf (p)pt =(1 αt)(1 βt), the complex absolute values of α and β are p.

Proof. By Lemma 3.1, the left hand side det(1−ϕpt : Vf,λ) is equal to det(1−Frpt : ⊗ Q

V(J1(N)p ) (f)λ). Thus it follows from Theorem 3.4.

Lemma 3.9 The map H1(X1(N), Q) → Hom(H1(X1(N), Q), Q) sending α to the linear form β → Tr(α ∩ wN β) is an isomorphism of T2(Γ1(N))-modules.

∗ Proof. It suﬃces to show T∗ ◦ w = w ◦ T . We deﬁnew ˜ : T1(N,n) → T1(N,n)by sending (E,P,C) → (E,Q,C) where E = E/(P +C), Q ∈ (E/C[N])/P ⊂E[N] is the inverse image of ζN by the isomorphism (P, ):(E/C[N])/P →μN where P ∈ E/C[N] is the image of P and C is the kernel of the dual of E/P →E. Then, ∗ we have s ◦ w˜ = w ◦ t, t ◦ w˜ = w ◦ s and hence T∗ ◦ w = w ◦ T .

3.4 Congruence relation

Let S be a scheme over Fp and E be an elliptic curve over S. The commutative diagram

E −−−FrE→ E ⏐ ⏐ ⏐ ⏐

S −−−FrS→ S → (p) × ∗ deﬁnes a map F : E E = E S FrS S called the Frobenius. The dual V = F : (p) → ◦ ◦ E E is called the Verschiebung. We have V F =[p]E,F V =[p]E(p) .

23 Lemma 3.10

− − ∗ det(1 Frpt : V(J1(N)p )) = det(1 p Frpt : V(J1(N) p )).

Proof. First, we show Fr◦ w = p◦w ◦ Fr. We have

Fr◦ w(E,P)=Fr(E/P ,Q)=(E(p)/P (p),Q(p)),

p◦w ◦ Fr(E,P)=p◦w(E(p),P(p))=(E(p)/P (p),pQ) (p) (p) (p) where Q ∈ E [N] is characterized by (P ,Q)N =(P, Q)N . Since (P ,Q )N = p (p) ◦ ◦ ◦ ◦ (P, Q)N =(P ,pQ)N , we have Fr w = p w Fr. Hence, we have w Fr = Fr◦p−1 ◦ w. ∗ By Lemma 3.9, it suﬃces to show that p Frp is the dual of Frp with respect to

∈ n the pairing α, wβ on VJ1(N)p .Forα, β J1(N) p [ ], we have

F∗α, wβ = w ◦ F∗α, β = (w ◦ F )∗α, β −1 ∗ = (Fr◦p ◦ w)∗α, β = α, wp∗F β and the assertion follows. Let N ≥ 1 be an integer and p N be a prime number. We deﬁne two maps

M →T a, b : 1(N)p 1(N,p) p by sending (E,P)to(E,P,F : E → E(p)) and to (E(p),P(p),V : E(p) → E) respectively. The compositions are given by s ◦ as◦ b id F = . (5) t ◦ at◦ b F p

M →T →

The maps a, b : 1(N)p 1(N,p) p induce closed immersions a, b : X1(N) p

T 1(N,p)p .

Proposition 3.11 Let N ≥ 1 be an integer and p N be a prime number. Then s, t : T 1(N,p) → X1(N) is ﬁnite ﬂat of degree p +1. The map

→ T

a b : X1(N)p X1(N) p 1(N,p) p is an isomorphism on a dense open subscheme.

→ T Proof. Since the maps a, b : X1(N)p 1(N,p) p are sections of projections

T →

1(N,p)p X1(N) p , they are closed immersions. Since both (1,F):X1(N) p

→ T →

X1(N)p X1(N) p and 1(N,p) p X1(N) p are ﬁnite ﬂat of degree p + 1, the assertion follows.

24 0 n Corollary 3.12 (cf. [12] Theorem 8.7.2) The canonical isomorphism Pic (X1(N))(Q)[ ] → 0 n Pic (X1(N))(Fp)[ ] makes the diagram

0 Q n −−−Tp→ 0 Q n Pic (X1(N⏐))( )[ ] Pic (X1(N⏐))( )[ ] ⏐ ⏐

∗ 0 n F∗+ p F 0 n Pic (X1(N))(Fp)[ ] −−−−−→ Pic (X1(N))(Fp)[ ] commutative.

Proof. By Proposition, we have a commutative diagram

∗ Tp=s∗t

unr −−−−→ unr Pic(X (N) ) Pic(X (N) ) 1⏐ p 1⏐ p ⏐ ⏐

∗ ∗ (t◦a)∗(s◦a) +(t◦b)∗(s◦b)

Pic(X (N) ) −−−−−−−−−−−−−−→ Pic(X (N) ) 1 p 1 p

∗ By (5), the bottom arrow is F∗ + pF . Proof of Theorem 3.7. By Corollary 3.12, we have

∗ 2 (1 − F∗t)(1 −pF t)=(1− Tpt + ppt ).

Taking the determinant, we get

∗ 2 2 det(1 − F∗t) det(1 −pF t)=(1− Tpt + ppt ) .

By Lemma 3.10, we get

2 det(1 − F∗t)=1− Tpt + ppt .

4 Construction of Galois representations: the case k>2

To cover the case k>2, we introduce a construction generalizing the torsion part of the Jacobian.

4.1 Etale cohomology In this subsection, we will recall very brieﬂy some basics on etale cohomology. For more detail, we refer to [8] [Arcata].

25 For a scheme X,anétale sheaf on the small étale site is a contravariant functor F : (Etale schemes/X) → (Sets) such that the map F → ∈ F ∗ ∗ F × ∈ (U) (si) (Ui) pr1(si)=pr2(sj)in (Ui U Uj) for i, j I i∈I → is a bijection for every family of étale morphisms (ϕi : Ui U)i∈I satisfying U = i∈I ϕi(Ui). An étale sheaf on X represented by a finite étale scheme over X is called locally constant. The abelian étale sheaves form an abelian category with enough injective objects. The étale cohomology Hq(X, ) is defined as the derived functor of the global section q functor Γ(X, ). For a morphism f : X → Y of schemes, the higher direct image R f∗ q Q Q ⊗ q Z nZ is defined as the derived functor of f∗. We write H (X, )= ←lim− nH (X, / ) q Q Q ⊗ q Z nZ and R f∗ = ←lim− nR f∗ / . Let f : X → S be a proper smooth morphism of relative dimention d and let F q be a locally constant sheaf on X. Then the higher direct image R f∗F is also locally constant for all q and is 0 unless 0 ≤ q ≤ 2d and its formation commutes with base change. More generally, assume f : X → S is proper smooth, U ⊂ X is the complement of a relative divisor D with normal crossings and F is a locally constant sheaf on U tamely ramified along D. Let j : U → X be the open immersion. Then, the higher q direct image R f∗j∗F is also locally constant and its formation commutes with base change ([8] Appendice 1.3.3 by L. Illusie to [Th. finitude]). If f : X → S is a proper smooth curve and if N is invertible on S, we have a 1 canonical isomorphism R f∗μN → JacX/S [N]. If S =Speck for a field k, the category of étale sheaves on S is equivalent to that F F of discrete sets with continuous Gk-actions by the functor sending to− lim→ L⊂k¯ (L). q For a scheme X over k, the higher direct image R f∗F is the étale cohomology group q F C H (Xk¯, ) with the canonical Gk-action. If k = , we have a canonical isomorphism q Z ⊗ Z Z → q Z Z

H (X, ) /N H (X, /N ) comparing the singular cohomology with the etale cohomology. Let X be a proper smooth variety over a ﬁeld k and f : X → X be an endomorphism. Then, for a prime number invertible in k, the Lefschetz trace formula ([8] [Cycle] Proposition 3.3) gives us

2 dimX − q ∗ q Q (Γf , ΔX )X×kX = ( 1) Tr(f : H (Xk¯, )). q=0

Assume k = Fp and apply the Lefschetz trace formula (loc. cit. Corollaire 3.8) to the iterates of the Frobenius endmorphism F : X → X. Then the zeta function is expressed by the determinant:

2 dimX − ∗ q Q (−1)q+1 Z(X, t)= det(1 F t : H (Xk¯, )) . q=0

26 ∗ q Theorem 4.1 (the Weil conjecture [9]) Let α be an eigenvalue of F on H (X¯, Q ). qk Then, α is an algebraic integer and its conjugates have complex absolute values p 2 .

4.2 Construction of Galois representations We brieﬂy sketch the construction in the higher weight case. For more detail, we refer to [7]. ≥ ≥ → Let N 5 and k 2. Proposition 3.5 is generalized as follows. Let f : EY1(N) Y1(N) be the universal elliptic curve and j : Y1(N) → X1(N) be the open immersion.

Proposition 4.2 There exists a canonical isomorphism

1 k−2 1 Q ⊗ R →

∗ ∗ É H (X1(N) ,j S R f ) Sk(Γ1(N))

of Tk(Γ1(N))Ê -modules.

1 k−2 1 Q

Corollary 4.3 H (X (N) ,j∗S R f∗ ) is a free T (Γ (N))É -module of rank 2.

1 É k 1

For a place λ| of Q(f), we put

1 k−2 1

V = H (X (N) ,j∗S R f∗Q ) ⊗ 1 Q(f) . É f,λ 1 É Tk(Γ (N)) λ

The Q(f)λ-vector space Vf,λ is a 2-dimensional -adic representation unramiﬁed at p N.

Theorem 4.4 The dual of the -adic representation Vf,λ is associated to f. Namely, for p N, we have

− −1 − k−1 2 det(1 ϕp t : Vf,λ)=1 ap(f)t + εf (p)p t .

k−1 2 Corollary 4.5 If we put 1 − ap(f)t + εf (p)p t =(1− αt)(1 − βt), the complex k−1 absolute values of α and β are p 2 .

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